Self-consistent cluster theory of the electron spectra of alloys

A method of self-consistent cluster expansion for the mass operator of the Green's function with allowance for short-and longrange order in alloys is proposed. A self-consistent system of equations for the coherent potential and the mass operator is obtained in the approximation in which the contributions of processes involving electron scattering by clusters of three or more atoms are ignored. The contributions of the scattering processes to the Green's function of the alloy decrease with increasing number of particles in a cluster and can be estimated by means of a certain small parameter. Analytic and numerical investigations of the energy dependence of the density of single-electron states of a binary alloy are made for different values of the parameter of the short-range order.Institute of Metal Physics, Ukrainian Academy of Sciences; Kiev University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 2, pp. 304–319, November, 1993.     

Theory of the conductivity of ordered alloys

A theory of the electrical conductivity of alloys going beyond the coherent potential approximation and taking into account statistical correlations in the scattering of the electrons by the atoms is developed. The two-particle Green's functions for the electrons is calculated with allowance for scattering by pairs of atoms (the zeroth approximation corresponds to the coherent potential approximation), the correlations being described by means of the parameters of the short-and long-range order. It is shown that the change in the electron spectrum on ordering leads to a significant change of the alloy conductivity.Institute of Metal Physics, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 2, pp. 279–293, May 1992.     

On the theory of high-temperature superconductivity

The construction of a theory of the superconductivity of oxide metals based on the polar model is discussed. The matrix electron Green's function is calculated with allowance for both the electron—phonon interaction and strong Coulomb correlations. The possible raising ofT _c due to the lattice instability associated with strong anharmonicity of the oxygen ion vibrations in perovskitelike compounds is considered. Strong electron correlations are taken into account in a one-bandt—J model. Because of a local restriction that precludes doubly occupied states in a system with strong correlation, onlyd-wave pairing is possible.N. N. Bogolyubov (deceased)The present paper was presented at the Fifth International Symposium on Selected Problems of Statistical Mechanics (August 22–24, 1989, Dubna) and is published here with small additions made without the participation of N. N. Bogolyubov.Joint Institute for Nuclear Research, 141980, Dubna, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 371–383, December, 1992.     

Static conductivity of a binary alloy in the traveling-cluster approximation

A study is made of the problem of finding the static electrical conductivity of a binary alloy in the tight-binding model with offdiagonal disorder in the presence of short-range order in the system. A general scheme is proposed for constructing selfconsistent approximations for averaged products of two Green's functions in order to determine the kinetic properties of disordered alloys. The approach corresponds to the traveling-cluster approximation used to find the electron spectra of binary alloys. The augmented-space formalism provides the framework for the treatment.Physicotechnical Institute, Ukrainian Academy of Sciences, Kharkov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 1, pp. 128–137, January, 1992.     

Kinetic equations in the method of two-time finite-temperature Green's functions. I. Renormalization of the collision integral

A system of equations that includes a generalized kinetic equation and equations for the static correlation functions is constructed for a normal quantum system of interacting Bose and Fermi particles with two-body interaction on the basis of the method of two-time finite-temperature Green's functions. The equations are in general valid for systems with arbitrary density of the particles. A method of successive approximation that makes it possible to go beyond the usual low-density expansion is discussed. The proposed method leads to a renormalization of the collision integral and makes it possible to obtain correlation functions for the total energy density, including its potential part.V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 3, pp. 351–372, September, 1993.     

The effects of geometrical parameters on force distributions and mechanics of carbon nanotubes: A critical study

In this paper, using the continuum approximation together with Lennard–Jones potential, a new semi-analytical expression is given to evaluate the van der Waals interaction between two single-walled carbon nanotubes. Based on this expression, two new formulations are also proposed to model multi-walled carbon nanotubes. In the first one, the interactions between each pair of shells from the inner and outer tubes are summed up over all of the pairs, whereas in the second formulation, a set of correction factors are applied to convert the results of double-walled carbon nanotubes to the correlated multi-walled ones. With respect to the present formulations, extensive studies on the variations of force distributions are performed by varying nanotube geometries so that the important features of the geometrical parameters are explored. Moreover, an acceptance condition for a nanotube at rest which is to be sucked into a semi-infinite nanotube is obtained. The influence of different geometrical parameters on the acceptance condition and suction energy, two main characteristics of nanotube-based systems for applications such as drug delivery and so on, is fully demonstrated. Lastly, an interesting relation for the maximum value of suction energy in terms of geometrical parameters is also extracted in this study.     

Fully polynomial-time approximation schemes for time–cost tradeoff problems in series–parallel project networks

We consider the deadline problem and budget problem of the nonlinear time–cost tradeoff project scheduling model in a series–parallel activity network. We develop fully polynomial-time approximation schemes for both problems using K-approximation sets and functions, together with series and parallel reductions.     

Bound states of a one-dimensional relativ1stic polaron in the single-loop approximation

A quasirelativistic two-zone model of a narrow-gap semiconductor is considered accounting with the interaction of the Dirac field of the electron with the classical dispersion-free phonon field in an external delta function impurity potential. Classical solutions corresponding to the discrete spectrum are found. In the single-loop approximation an analysis is given of the effect of polarization of the Fermion vacuum by the phonon field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 51–54, 1982.In conclusion, the authors express their gratitude to P. P. Kulish and V. N. Popov for useful discussions.     

Equivalent formulations and necessary optimality conditions for the Lennard–Jones problem

The minimization of molecular potential energy functions is one of the most challenging, unsolved nonconvex global optimization problems and plays an important role in the determination of stable states of certain classes of molecular clusters and proteins. In this paper, some equivalent formulations and necessary optimality conditions for the minimization of the Lennard–Jones potential energy function are presented. A new strategy, the code partition algorithm, which is based on a bilevel optimization formulation, is proposed for searching for an extremal Lennard–Jones code. The convergence of the code partition algorithm is proved and some computational results are reported.     

A numerical method for the Cahn–Hilliard equation with a variable mobility

We consider a conservative nonlinear multigrid method for the Cahn–Hilliard equation with a variable mobility of a model for phase separation in a binary mixture. The method uses the standard finite difference approximation in spatial discretization and the Crank–Nicholson semi-implicit scheme in temporal discretization. And the resulting discretized equations are solved by an efficient nonlinear multigrid method. The continuous problem has the conservation of mass and the decrease of the total energy. It is proved that these properties hold for the discrete problem. Also, we show the proposed scheme has a second-order convergence in space and time numerically. For numerical experiments, we investigate the effects of a variable mobility.     

Superconductivity and Ferromagnetism in Nonadiabatic Systems with Magnetic Impurity: A Step Beyond the Migdal Theorem

The possibility of the ferromagnetic ordering of a paramagnetic impurity in nonadiabatic superconducting systems is investigated. The effect of the relative shift of the Fermi surface by the internal magnetic field, the exchange interaction of the impurity ions through the conductivity electrons, and the spin–orbit interaction of the nonmagnetic impurity are taken into account. The problem is solved in the linear approximation with respect to the nonadiabaticity by taking the vertex and crossing diagrams corresponding to the electron–phonon and the electron–impurity interactions into account. We obtain basic equations of the superconductivity theory for nonadiabatic systems with the ferromagnetic ordering of the impurity spins and show that the nonadiabaticity alters the superconducting transition temperature T _c and the critical impurity concentration. The behavior of the magnetic-ordering temperature T _C as a function of the impurity concentration c in the superconductive state of the nonadiabatic system is also investigated. We obtain the phase diagram (T,c) and show that the nonadiabaticity effects lead to the enlargement of the domain where the superconductivity and the ferromagnetism exist simultaneously.     

Method of functional integration and an eikonal approximation for potential scattering amplitudes

Conclusions We have obtained an exact closed expression for the potential scattering amplitude of particles with spin o and 1/2 as a functional integral with respect to trajectories. This has made possible a relatively simple expansion of the amplitude in powers of the small parameter 1/E. The first term of the expansion is an eikonal approximation for the amplitude for scattering through any angle and in the case of dynamically small angles (pR)^–1/2 is identical with the Glauber representation.We have found the asymptotic form of the scattering amplitude for two particles that exchange virtual mesons. A similar result was obtained in [7] by functional integration with respect to the external fields of the exact Green's functions and a subsequent eikonal expansion on the mass shell. The equivalence of this more accurate method to the approximation described in the present paper (see also [8, 9]) is connected with the interesting problem of the commutativity of the operations of eikonal approximation and second quatization.If the latter do commute, the Glauber representation for the amplitude (18) in quantum field theory is a consequence of the eikonal approximation in quantum mechanics.Joint Institute of Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 4, No. 1, pp. 22–32, July, 1970.     

Composite generalized Laguerre–Legendre pseudospectral method for Fokker–Planck equation in an infinite channel

In this paper, we propose a composite generalized Laguerre–Legendre pseudospectral method for the Fokker–Planck equation in an infinite channel, which behaves like a parabolic equation in one direction, and behaves like a hyperbolic equation in other direction. We establish some approximation results on the composite generalized Laguerre–Legendre–Gauss–Radau interpolation, with which the convergence of proposed composite scheme follows. An efficient implementation is provided. Numerical results show the spectral accuracy in space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are also very appropriate for many other problems on multiple-dimensional unbounded domains, which are not of standard types.     

On the orthogonality of the MacDonald's functions

A proof of an orthogonality relation for the MacDonald's functions with identical arguments but unequal complex lower indices is presented. The orthogonality is derived first via a heuristic approach based on the Mehler–Fock integral transform of the MacDonald's functions, and then proved rigorously using a polynomial approximation procedure.     

Existence and global bounds for a fluid model of plasma display technology

This article considers the fluid model for the discharge of plasma particle species in display technology. The fluid equations are coupled with Poisson's equation, which describes the effect of the charged particles on the electric field. The diffusion and mobility coefficients for the positive ion particles depend on the electric field, while those for the electrons depend on the electron mean energy. The reaction rates are proportional to the products of the densities of the reacting particles involved in the particular ionization, conversion or recombination reactions. Moreover, the ionization coefficients are dependent on the electric field, which varies spatially and temporally. The main ionization and discharge reactions are described by an initial-boundary value problem for a system of coupled parabolic–elliptic partial differential equations. The system is first analyzed by upper–lower solution method. By means of the a priori bounds obtained for an arbitrary time, the existence of solution for the initial-boundary value problem is proved in an appropriate Hölder space.     

Full asymptotics of spline Petrov–Galerkin methods for some periodic pseudodifferential equations

In this paper we study the existence of a formal series expansion of the error of spline Petrov–Galerkin methods applied to a class of periodic pseudodifferential equations. From this expansion we derive some new superconvergence results as well as alternative proofs of already known weak norm optimal convergence results. As part of the analysis the approximation of integrals of smooth functions multiplied by splines by rectangular rules is analyzed in detail. Finally, some numerical experiments are given to illustrate the applicability of Richardson extrapolation as a means of accelerating the convergence of the methods.     

An Iterative Global Optimization Algorithm for Potential Energy Minimization

In this paper we propose an algorithm for the minimization of potential energy functions. The new algorithm is based on the differential evolution algorithm of Storn and Price (Journal of Global Optimization, vol. 11, pp. 341–359, 1997). The algorithm is tested on two different potential energy functions. The first function is the Lennard Jones energy function and the second function is the many-body potential energy function of Tersoff (Physics Review B, vol. 37, pp. 6991–7000, 1988; vol. 38, pp. 9902–9905, 1988). The first problem is a pair potential and the second problem is a semi-empirical many-body potential energy function considered for silicon-silicon atomic interactions. The minimum binding energies of up to 30 atoms are reported.Visitor at the Institute for Mathematics and its Applications, University of Minnesota, USA.     

Inégalités de Faber–Krahn et Inclusion de Sobolev–Orlicz

Equivalence is demonstrated between a generalized form of the inequality of Faber–Krahn and an inequality of Sobolev–Orlicz. Equivalences with estimates on the decay of the heat kernel and inequalities on capacities and Green's functions are also given.     

Specific heat and mean energy of a Heisenberg ferromagnet in the Green's function method

A system of equations that permits calculation of the longitudinal correlation function in terms of the correlation functions of the transverse spin components is proposed. In the Green's function method, the system permits calculation of the susceptibility, mean energy, and specific heat in the same approximation as the magnetization. The experimental data on the specific heat in Gd and EuS agree qualitatively with the conclusions of the theory.Ural State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 127–134, April, 1993.